On first subharmonic bifurcations in a branch of Stokes waves

TL;DR

This paper analyzes bifurcations in Stokes water waves, revealing the existence and structure of M-subharmonic bifurcations and their relation to the primary Stokes bifurcation point.

Contribution

It provides a detailed description of M-subharmonic bifurcations, their occurrence after the first Stokes bifurcation, and the structure of the connected set of solutions.

Findings

No M-subharmonic bifurcations before the first Stokes bifurcation.

M-subharmonic bifurcation points exist after the first Stokes bifurcation for large M.

The set of bifurcating solutions forms a closed connected continuum.

Abstract

Steady surface waves in a two-dimensional channel are considered. We study bifurcations, which occur on a branch of Stokes water waves starting from a uniform stream solution. Two types of bifurcations are considered: bifurcations in the class of Stokes waves (Stokes bifurcation) and bifurcations in a class of periodic waves with the period M times the period of the Stokes wave (M-subharmonic bifurcation). If we consider the first Stokes bifurcation point then there are no M-subharmonic bifurcations before this point and there exists M-subharmonic bifurcation points after the first Stokes bifurcation for sufficiently large M, which approach the Stokes bifurcation point when M tends to infinity. Moreover the set of M- subharmonic bifurcating solutions is a closed connected continuum. We give also a more detailed description of this connected set in terms of the set of its limit points, which must contain extreme waves, or overhanging waves, or solitary waves or waves with stagnation on the bottom, or Stokes bifurcation points different from the initial one.